Genericity of hyperbolic homogeneous vector fields in R3

construction of vector fields with positive lyapunov exponents

in this thesis our aim is to construct vector field in r3 for which the corresponding one-dimensional maps have certain discontinuities. two kinds of vector fields are considered, the first the lorenz vector field, and the second originally introced here. the latter have chaotic behavior and motivate a class of one-parameter families of maps which have positive lyapunov exponents for an open in...

Polynomial Normal Forms for Vector Fields on R3

The present paper is devoted to studying a class of smoothly (C∞) finitely determined vector fields on R3. Given any such generic local system of the form ẋ = Ax + · · · , where A is a 3 × 3 matrix, we find the minimal possible number i(A) such that the vector field is i(A)-jet determined, and we find the number μ(A) of moduli in the C∞ classification. We also give a list of the simplest normal...

Genericity of Blackhole Formation in the Gravitational Collapse of Homogeneous Self-interacting Scalar Fields

The gravitational collapse of a wide class of self-interacting homogeneous scalar fields models is analyzed. The class is characterized by certain general conditions on the scalar field potential, which, in particular, include both asymptotically polynomial and exponential behaviors. Within this class, we show that the generic evolution is always divergent in a finite time, and then make use of...

Linearization of families of germs of hyperbolic vector fields

On Algebraic Proofs of Stability for Homogeneous Vector Fields

We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sums of squares certificates and hence such a Lyapunov fun...